According to the Encyclopedia of Photography, "The luminance of an extended source or surface is invariant with viewing distance. Luminance is also invariant within a lossless optical system because changes in image size are balanced by inverse changes in solid angle. The luminance of a Lambertian diffuser is invariant with viewing angle because the reduced power reflected off-axis is balanced by a reduction in projected area."

Do that if you want to average the light that is falling on both sides of the subject (i.e., from the area covered by the dome). But when using light that is uneven, why would you want to average it as a matter of course? Sometimes you do, yes. But not always; not even often, I would say, for my own pix. Do this in deliberate ratio lighting, e.g. a side-lit portrait, and you overexpose. The more contrasty the lighting ratio, the more overexposed you will be.

Point it at the light for which you want to correctly expose - the "main light." You have gone to all this trouble to craft light, or choose a location and time of day, that will sculpt the subject the way you want it, why would you then average that light with the dark side, which you have intentionally made dark? You crafted, or chose to use, the light that way because you want the dark side to be dark. If you don't want it to be dark, then change the fill ratio. Don't average the exposures for the light side and the dark side as a matter of course; it does not make sense. For best results, one meters the main light source, unless in very flat, even light, in which case one could point the dome practically anywhere and get the same reading.

Y' pays y'er money and y'er takes y'er choice.

Yes, there ARE areas of the scene that I would want "dark", but the question is "How dark?"

If you measure ONLY for the Main, you wil get ... a density in the negative that correlates to 18% gray, and the "dark" areas will sink into some level of unmetered black.
All well and good, if you are a devotee of "It ain't good unless there is a *really* dense, black area".

It depends .. on the situation, the intent, the aesthetics (aesthetics - there is a cop-out).

Generally (read: exceptions would not be surprising) I would use out-of-the-box `Set the meter and dome to "Incident": At the subject, point the meter more or less at the camera' and blaze away.

We do!
There is less light, reflected off the piano, reaching the camera than at 150 m, or 10 m.
But don't forget that the image of that piano will be proportionally smaller.
As mentioned before: both image size and light intensity follow the same geometry. So though there's less light, that light has to fill in a smaller spot on film. The intensity per area unit will be the same at 300 m as it would be at 150 m, or 10 m.
So the setting to use is the same too.

If at 300 m we use a tele lens and we fill the image with the piano, a lesser quantity of light fills the entire image, so we have a different exposure.

If I place on the piano a grey card and read it with a spot reflected meter, which only reads light reflected from the card (that is, supposing the card "fills" the reading angle of the spot lightmeter), my understand of physics tells me that my spot meter will give me a different exposure than the incident light meter used near the piano.

I am not convinced that my understanding of physics is right, though. Actually I very much doubt it.

If I take a picture of a lit monument at night, the exposure for the monument isn't the same even if I am far from the monument?

According to the Encyclopedia of Photography, "The luminance of an extended source or surface is invariant with viewing distance. Luminance is also invariant within a lossless optical system because changes in image size are balanced by inverse changes in solid angle. The luminance of a Lambertian diffuser is invariant with viewing angle because the reduced power reflected off-axis is balanced by a reduction in projected area."

If at 300 m we use a tele lens and we fill the image with the piano, a lesser quantity of light fills the entire image, so we have a different exposure.

Correct. The longer lens will change the size of the image, compared to, say, a shorter lens, but cannot change the amount of light it receives form the subject. So telelenses project a larger, but darker image, i.e. are slower, than shorter lenses.

(Assuming the same size light gathering area, of course. You can give a telelens a larger front lens/entrance pupil, and it will be able to gather more light than a smaller, shorter lens. It will need that - collect more light - to be able to project a larger image that is as bright as the smaller image. Make the front lens large enough, and it will be able to collect as much light as is needed to project a lager image that is also brighter than that of the shorter lens. But that shorter lens can also be made faster, etc., etc.)

Originally Posted by Diapositivo

If I place on the piano a grey card and read it with a spot reflected meter, which only reads light reflected from the card (that is, supposing the card "fills" the reading angle of the spot lightmeter), my understand of physics tells me that my spot meter will give me a different exposure than the incident light meter used near the piano.

Assuming that there are no other variables, it does not.
Why would it?

Originally Posted by Diapositivo

If I take a picture of a lit monument at night, the exposure for the monument isn't the same even if I am far from the monument.

I'm really puzzled.

The answer is in what is said before, and what was quoted from that encyclopedia.

The question is why, if you measure at your subject (which you certainly should with a light two feet away from the subject), will the reading stil be correct if you set the cmaera up two miles from your subject.
The light seen by the camera has to travel those two miles after being reflected off the subject.

So before anyone else gives the same non-answer you gave, tell us why you can use the same setting you should use with the camera not two miles, but six feet from your subject.

Incident metering measures the light falling on the object, and that does not change no matter where the camera, or eye, is located. A face, for example, properly exposed at one meter, will remain properly exposed at two hundred meters, due to the fact that its image size decreases - proportionately.

A star many light years away is just as bright, per unit area, as it is from a distance of one meter.

Incident metering measures the light falling on the object, and that does not change no matter where the camera, or eye, is located.

I don't understand why people feel the need to harp on that point.
It's not what is being discussed,. Not what the question is about.

I'll tell you one more time: that "light falling on the object" has to travel from that object to your camera for you to be able to capture that object on film. The question is why there is no light loss due to that.
The answer is that there is.

Originally Posted by Ed Sukach

A face, for example, properly exposed at one meter, will remain properly exposed at two hundred meters, due to the fact that its image size decreases - proportionately.

Yes.
I have told you that a couple of times now...

Originally Posted by Ed Sukach

A star many light years away is just as bright, per unit area, as it is from a distance of one meter.

You have been talking about sun as a source of light for this discussion. That's true SUN is far far far away making the little distance from subject to camera irrelavant. How do you explain, the same rule holds true when the light source is a studio flash/light? You, too, can take incident reading at the subject. Now, the light source, and point source at that, is only 10 feet away. You, the camera man/woman can be at 6 feet, 10 feet, or 20 feet, and incident reading is still valid.

The only rationale I can come up with is that the point source light falls on the subject. In this journey of the light, it spreads as it travels. Once the subject is illuminated, the reflected light doesn't spread. I don't know why and can't explain why.

Image size? OK, but the size of subject being 10% of the total frame or 100%, say just the cheek of the person or the whole face, at either magnifications, the spot in question, (let's say the cheek) is exposed identically. Does the size really matter??

The only rationale I can come up with is that the point source light falls on the subject. In this journey of the light, it spreads as it travels. Once the subject is illuminated, the reflected light doesn't spread. I don't know why and can't explain why.

You will not have to search for an explanation why it would not spread, because it indeed does spread.
Read the thread. All is revealed. Several times, in fact.

Just for clarity, I had written a sentence without a question mark, I corrected it immediately after posting but Q.G. had already answered to my post. My statement

If I take a picture of a lit monument at night, the exposure for the monument isn't the same even if I am far from the monument.

Should read:

If I take a picture of a lit monument at night, the exposure for the monument isn't the same even if I am far from the monument?

The explanation given is not clear to me. When I am far, the angle at which I see the piano is narrower than when I am near. The piano goes on spreading light in all directions, but when I am far I am reached by only one smaller quota of that light.

When I am near, I am reached by a larger quota.

So, the explanation confuses me even more.

When I am far, not just the light has "diverged more" than when I am near (it is more "spreaded" just like butter is thinner if spread on a larger slice of bread) but, in addiction to this, I even see a smaller "angle" of it.

I visualize light as if it were the surface of a rubber balloon. The more I inflate the balloon, the more the same rubber is spread over a larger surface, so the quantity of rubber per unit of surface is smaller.

If I consider the solid angle, well, even inflating the balloon, the amount of rubber for a certain solid angle never changes.

But when I take a picture of the piano from far, with a tele lens, I am using just a smaller "solid angle" of the light reflected by the piano.